Implement conversion from laurent series to rational function field

Author: user202729

src/sage/rings/fraction_field.py

@@ -557,6 +557,30 @@ def is_exact(self):
         """
         return self.ring().is_exact()
 
+    def _convert_from_finite_precision_laurent_series(self, x):
+        """
+        Construct an element of this fraction field approximating a Laurent series.
+
+        INPUT:
+
+        - ``x`` -- a Laurent series, must have finite precision
+
+        OUTPUT: Element of ``self``
+
+        This internal method should not be used directly, use :meth:`__call__` instead,
+        which will delegates to :meth:`_element_constructor_`. There are some tests there.
+
+        .. NOTE::
+
+            Uses the algorithm described in `<https://mathoverflow.net/a/14874>`_.
+            This may be changed to use Berlekamp--Massey algorithm or something else
+            to compute Padé approximant in the future.
+        """
+        integral_part, fractional_part = self(x.truncate(1)), x.truncate_neg(1)
+        if fractional_part.is_zero():
+            return integral_part
+        return integral_part + ~self._convert_from_finite_precision_laurent_series(~fractional_part)
+
     def _element_constructor_(self, x, y=None, coerce=True):
         """
         Construct an element of this fraction field.
@@ -662,6 +686,9 @@ def _element_constructor_(self, x, y=None, coerce=True):
             sage: f.parent()
             Power Series Ring in x over Rational Field
             sage: F(f)
+            doctest:warning...
+            DeprecationWarning: Conversion from power series to rational function field is deprecated, use .truncate() instead
+            See https://github.com/sagemath/sage/issues/39485 for details.
             -x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
 
         Conversion from Laurent series to rational function field gives an approximation::
@@ -672,48 +699,56 @@ def _element_constructor_(self, x, y=None, coerce=True):
             sage: f.parent()
             Laurent Series Ring in x over Rational Field
             sage: F(f)
-            Traceback (most recent call last):
-            ...
-            TypeError: cannot convert 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19 + O(x^20)/1 to an element of Fraction Field of Univariate Polynomial Ring in x over Rational Field
+            1/(x + 1)
             sage: f = f.truncate(20); f  # infinite precision
             1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19
             sage: f.parent()
             Laurent Series Ring in x over Rational Field
             sage: F(f)
-            Traceback (most recent call last):
-            ...
-            TypeError: cannot convert 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19/1 to an element of Fraction Field of Univariate Polynomial Ring in x over Rational Field
+            -x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
             sage: f = 1/(x*(x+1))
             sage: f.parent()
             Laurent Series Ring in x over Rational Field
             sage: F(f)
-            Traceback (most recent call last):
-            ...
-            TypeError: cannot convert x^-1 - 1 + x - x^2 + x^3 - x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - x^12 + x^13 - x^14 + x^15 - x^16 + x^17 - x^18 + O(x^19)/1 to an element of Fraction Field of Univariate Polynomial Ring in x over Rational Field
+            1/(x^2 + x)
 
         ::
 
             sage: K.<x> = FunctionField(QQ)
             sage: R.<x> = QQ[[]]
             sage: f = 1/(x+1)
             sage: K(f)
+            doctest:warning...
+            DeprecationWarning: Conversion from power series to rational function field is deprecated, use .truncate() instead
+            See https://github.com/sagemath/sage/issues/39485 for details.
             -x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
             sage: f = Frac(R)(1/(x+1))
             sage: K(f)
-            Traceback (most recent call last):
-            ...
-            TypeError: cannot convert 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 - x^13 + x^14 - x^15 + x^16 - x^17 + x^18 - x^19 + O(x^20)/1 to an element of Fraction Field of Univariate Polynomial Ring in x over Rational Field
+            1/(x + 1)
             sage: f = 1/(x*(x+1))
             sage: K(f)
-            Traceback (most recent call last):
-            ...
-            TypeError: cannot convert x^-1 - 1 + x - x^2 + x^3 - x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - x^12 + x^13 - x^14 + x^15 - x^16 + x^17 - x^18 + O(x^19)/1 to an element of Fraction Field of Univariate Polynomial Ring in x over Rational Field
+            1/(x^2 + x)
         """
         if isinstance(x, (list, tuple)) and len(x) == 1:
             x = x[0]
         if y is None:
             if parent(x) is self:
                 return x
+            from sage.rings.polynomial.polynomial_ring import PolynomialRing_generic
+            if isinstance(self.ring(), PolynomialRing_generic):
+                from sage.rings.power_series_ring_element import PowerSeries
+                from sage.rings.laurent_series_ring_element import LaurentSeries
+                if isinstance(x, PowerSeries):
+                    from sage.misc.superseded import deprecation
+                    deprecation(
+                        39485,
+                        "Conversion from power series to rational function field is deprecated, use .truncate() instead",
+                    )
+                if isinstance(x, LaurentSeries):
+                    from sage.rings.infinity import infinity
+                    if x.prec() == infinity:
+                        return self(x.laurent_polynomial())
+                    return self._convert_from_finite_precision_laurent_series(x)
             ring_one = self.ring().one()
             try:
                 return self._element_class(self, x, ring_one, coerce=coerce)